The $n$-th Mersenne number is $M_n=2^n-1$. Let $A$ be the set of squarefree positive integers $a$ such that $M_n=a b^2$ for some positive integers $n$, $b$. My question is regarding the natural density of $A$, defined as $$ \delta_A=\lim_{X \rightarrow \infty} \frac{\# \{a \in A | a \le X\}}{X}. $$

Question: Show that $\delta_A=0$.